Extracting algebraic constraints from the input data
I'm assuming the question intends to find the maximally constraining bounding relation from among a large variety of other possible relations.

The bounding relation is likely linked to the chaos idea of attractors$^1$. Some contemporary visualizations of attractors in phase space are visualized as bounds rather than translucent paths or cross-sectional means.

The simple form used as an example, the circle, never actually exist in nature. It is an approximation of a differential equation for an orbit with a eccentricity of zero in three space projected onto a parallel plane. But even if this problem is restricted to $\mathbb{R}^2$, the concepts of attractors may pertain.

If quadratic or cubic approximation is adequate for your purposes then it is a matter of finding the zero terms in the generalization. To find non-polynomial relations, the differential equations are the first things to find, but there may be no equivalent closed form in algebra

[1] http://ijfritz.byethost4.com/Chaos/ch_close.htm?i=1